Research On Steady-state Properties Of Polydisperse Granular Mixtures

In this paper, we studied steady-state properties of polydisperse granular gases in one and two dimension. The steady-state properties include dynamical behaviors, granular energies and granular stresses, et al. We showed that the distribution of velocities for dissipative gases depend only on two parameters. Finally, we presented the zeroth law of non-equilibrium thermodynamics.First, the particles employed in our study are assigned diameters representative of a power-law size distribution according to experiment of granularity measurement and the fractal theory. The width of size distribution of the mixture is characterized by the only parameter the fractal dimension D . It can be considered as a measurement of the inhomogeneity of the size distribution. With the increasing value of D, the size distribution of particles is widened. We presented an model of 1D power-law granular mixtures. By Monte Carlo method, we simulate the dynamic properties of the mixture. Non-Gaussian velocity distribution and clusterization arise. The deviation and clusterization become more and more pronounced as the fractal dimension D increases. Furthermore, we proposed the definition of the partial and global granular temperatures. The partial granular temperature represents the mean kinetic energy per particle of i th component. The global granular temperature is the statistic average value of the partial granular temperatures of n components and also indicates that the assembly achieves asymptotically a statistical steady state, as the energy dissipation is balanced by the energy injection. The global granular temperature reduces and average dissipated energy per particle increases with the augment of D .Second, we presented a two-dimensional model of polydisperse granular mixtures with a power-law size distribution. By numerical simulation, we investigated the effect of the power-law size distribution on the velocity distributions, partial distributions, mean collision time and collision rate of the system. We found that the velocity distribution, the partial distribution, the mean collision time and the collision rate are prominently influenced by the size distribution of the system, i.e., the fractal dimension D . Moreover, we studies the properties of granular energy and granular pressures of the power-law mixture. The global granular temperature reduces and average dissipated energy per particle increases as the width of the size distribution widen. The larger particles tend to have more granular energy than smaller particles, and that this difference in granular energy increases with the difference in size between the two particle types. The pressures in the power-law system are found to decrease as the width of the particle size distribution is increased. In particular, the smaller particles produce relatively less pressures and larger particles produce relatively more pressures for both collisional and kinetic contributions to the pressure tensor. This difference in pressures increases with the difference in size between the two particle types.Third, we compared the velocity distributions and partial distributions of granular gases which were driven by uniform heating and boundary heating. We study the behavior of the velocity distributions of particles as a function of the width of size distribution D , the area fractionφ, the total particle number N and the restitution coefficientη. There are striking differences between uniform heating and boundary heating. The form of the velocity distribution for uniform and boundary heating is primarily controlled by two parametersηand the heating-dissipation ratio q . Furthermore, we show that there is no existence for a universal velocity distribution with an exponentα=1.5. Instead, a family of distributions with exponents covering a range of values 0.6≤α2<2. The values of exponentsα2 are depend on the parameters D ,η,φand N .Finally, based on the set up of the concept of steady state in non-equilibrium system, we propose the concept of global granular temperature and further the zeroth law of non-equilibrium granular systems: when two granular systems with the same size distribution, undergoing the same heating and dissipation mechanism, and possessing the same global granular temperatures, are brought into thermal contact, they do not exchange energy macroscopically and the granular temperatures of the two subsystems are unaltered. The overall system comprised of these two subsystems state in the steady state and the global temperature of the overall system does not change. It should be stressed that the non-equilibrium zeroth law proposed in this paper has two provisos, the same heating and dissipation mechanisms. The universality of the zeroth law of non-equilibrium thermodynamics is much smaller than the zeroth law of thermodynamics because of the complexities of non-equilibrium systems reactions.